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In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary finite dimension. In this setting, the orientation of an ordered basis is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple rotation. Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a rotation alone, but it is possible to do so by reflecting the figure in a mirror. As a result, in the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral). The orientation on a real vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called . ==Definition== Let ''V'' be a finite-dimensional real vector space and let ''b''1 and ''b''2 be two ordered bases for ''V''. It is a standard result in linear algebra that there exists a unique linear transformation ''A'' : ''V'' → ''V'' that takes ''b''1 to ''b''2. The bases ''b''1 and ''b''2 are said to have the ''same orientation'' (or be consistently oriented) if ''A'' has positive determinant; otherwise they have ''opposite orientations''. The property of having the same orientation defines an equivalence relation on the set of all ordered bases for ''V''. If ''V'' is non-zero, there are precisely two equivalence classes determined by this relation. An orientation on ''V'' is an assignment of +1 to one equivalence class and −1 to the other.〔Rowland, Todd. "Vector Space Orientation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/VectorSpaceOrientation.html〕 Every ordered basis lives in one equivalence class or another. Thus any choice of a privileged ordered basis for ''V'' determines an orientation: the orientation class of the privileged basis is declared to be positive. For example, the standard basis on R''n'' provides a standard orientation on R''n'' (in turn, the orientation of the standard basis depends on the orientation of the Cartesian coordinate system on which it is built). Any choice of a linear isomorphism between ''V'' and R''n'' will then provide an orientation on ''V''. The ordering of elements in a basis is crucial. Two bases with a different ordering will differ by some permutation. They will have the same/opposite orientations according to whether the signature of this permutation is ±1. This is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Similarly, let ''A'' be a nonsingular linear mapping of vector space R''n'' to R''n''. This mapping is orientation-preserving if its determinant is positive.〔 Weisstein, Eric W. "Orientation-Preserving." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Orientation-Preserving.html〕 For instance, in R3 a rotation around the ''Z'' Cartesian axis by an angle ''α'' is orientation-preserving: :: while a reflection by the ''XY'' Cartesian plane is not orientation-preserving: :: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Orientation (vector space)」の詳細全文を読む スポンサード リンク
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